AN IMPLICIT BLOCK JACOBI APPROACH FOR HIGH-ORDER FLUX RECONSTRUCTION METHOD

High-order methods are quickly becoming popular for turbulent flows as the amount of computer processing power increases. The flux reconstruction (FR) method presents a unifying framework for a wide class of high-order methods including discontinuous Galerkin (DG), Spectral Difference (SD), and Spectral Volume (SV). It offers a simple, efficient, and easy way to implement nodal-based methods that are derived via the differential form of the governing equations. Whereas high-order methods have enjoyed recent success, they have been known to introduce numerical instabilities due to polynomial aliasing when applied to under-resolved nonlinear problems. Aliasing errors have been extensively studied in reference to DG methods; however, their study regarding FR methods has mostly been limited to the selection of the nodal points used within each cell. Here, we extend some of the de-aliasing techniques used for DG methods, primarily over-integration, to the FR framework. Our results show that over-integration does remove aliasing errors but may not remove all instabilities caused by insufficient resolution (for FR as well as DG).

In this paper, the ability of high-order flux reconstruction numerical schemes to perform accurate and stable computations of compressible turbulent flows on coarse meshes is investigated. Two new flux reconstruction schemes, which are optimized for wave dissipation and dispersion properties, are compared to the nodal discontinuous Galerkin and spectral difference methods recovered via the energy-stable flux reconstruction method. The Taylor–Green vortex benchmark problem at is used as a simple a priori test of the numerics. Dissipation rates computed from kinetic energy, vorticity, and pressure dilatation are plotted against reference solutions. Results show that, at low mesh resolution, the flux reconstruction schemes are highly accurate across a range of orders of accuracy, although oscillations can appear in the solution at orders of six and above. Although the flux reconstruction method has a built-in stabilization mechanism, an additional means of damping these instabilities is required. The schemes vary in the amount of numerical dissipation and resolution of the turbulent spectrum. One of the optimized flux reconstruction schemes (the optimized flux reconstruction scheme) is shown to have greater spectral accuracy than any of the others tested, motivating its future usage for high-order high-fidelity computational fluid dynamics.

A high-order accurate flow solution method for quadrilateral elements is developed and tested. The method employs the flux reconstruction (FR) approach proposed by Huynh. Ideal orders of accuracy are achieved for several problems on structured-based grids while some deterioration is observed on purely unstructured grids. Cares for curved wall boundary are also shown. The localized artificial diffusivity (LAD) method is incorporated to capture discontinuities that is crucial for compressible flow simulations. The FR+LAD works very well for 1D test cases but multi-dimensional shock problems requires further modifications to confine the artificial diffusivity to the shock location. Hybrid scheme of the FR and finite difference method taking advantage of 1D-feature alleviates the instability. I. Introduction Expanding demands for the computational fluid dynamics (CFD) requires more reliable numerical schemes. Specifically, turbulent flow simulations, acoustic simulation, combustion simulations and their combined problems. The key words in such simulations may be high accuracy and high resolution. In structured grid CFD, a compact finite difference scheme satisfies both requirements at higher level than existing other methods. If we put more emphasis on easier handling of complex geometries and flow-data dependent grid generation (grid adaptation), then unstructured grid method should also be explored and it should depart from 2nd-order scheme with limiter. Recent efforts in this area are development of the discontinuous Galerkin (DG) method, 1,2 the spectral finite volume (SV) method 3 and the staggered-grid multi-domain method spectral finite difference (SG) method 4,5 and its evolutional method called a spectral difference method. 6 The common and maybe the only way for those unstructured methods is to add degrees of freedom in each cell to achieve high order since the wide stencil of the original grid is not possible due to unstructured nature. The ENO-type method that uses original degrees of freedom in the grids and extends adaptive stencils to the flow solutions is also explored but the difficulty of extending stencils due to undirectional nature of unstructured grids and the unsuitability for parallel computing prohibits wide applications. A flux reconstruction (FR) scheme 7,8 studied in the paper is the new one among the former group. It uses differential form of conservation laws like the SD method but it is more general and includes equivalent counter part of the schemes in the group. In this paper, the performance of the FR is tested by a newly developed Navier-Stokes solver. Realistic flow applications of the FR in the Navier-Stokes equations are seldom seen since it is relatively new scheme. The Localized artificial diffusivity (LAD) 9–13 is incorporated for the shock capturing in the FR. The last item is motivated by the challenge and success of combining the SD and LAD in the recent study. 11,12 Wang 14 developed the lifting collocation penalty (LCP) method to extend the FR to triangular or tetrahedral grids and applied the LCP-FR combined method for hybrid unstructured grids. In the present study, only quadrilateral cells are treated to directly apply the LAD method that was developed on structured grids. The shock capturing by the FR+LAD high order scheme is tried.

Center-to-face momentum interpolation and face-to-center flux reconstruction in Euler-Euler simulation of gas-solid flows

We propose a high-order immersed boundary (IB) method for simulating inviscid flows using the flux reconstruction (FR) method, which offers the advantages of a compact stencil, high-order near-wall accuracy, and the ability to handle thin geometries and sharp corners. We utilize the FR method on a Cartesian grid to achieve a compact stencil, which enhances accuracy without requiring surrounding cell information, laying the foundation for future adaptive mesh refinement. Compared to the symmetry wall condition treatment, we combine the curvature corrected symmetry technique with the FR-based IB method to achieve a high-order accuracy near the wall boundary and utilize a third-order polynomial distribution to impose the boundary conditions. We extend the multi-valued ghost-cell method to the present approach to handle thin geometries and sharp corners in the ghost-cell-based IB method. The proposed method can be easily extended to three-dimensional cases. The numerical framework is validated through several two-dimensional simulations of inviscid flows. The approach is first demonstrated for a superposition of an isentropic vortex and a background flow problem with a Mach number of 0.2, which examines the order of accuracy without wall effects in the grid convergence study. Then, inviscid flow around a cylinder and flow through a channel with a bump are tested and compared to body-fitted simulations. Finally, flows around a National Advisory Committee for Aeronautics 0012 airfoil with zero and non-zero angles of attack are demonstrated to show the robustness for the thin geometry calculation. In particular, third-order accuracy is maintained in the near-wall region.

Immersed boundary method for high-order flux reconstruction based on volume penalization

The immersed boundary (IB) method is an approach to fluid-structure interaction that uses Lagrangian variables to describe the structure and Eulerian variables to describe the fluid. Explicit time stepping schemes for the IB method require solvers only for Eulerian equations, for which fast Cartesian grid solution methods are available. Such methods are relatively straightforward to develop and are widely used in practice but often require very small time steps to maintain stability. Implicit-time IB methods permit the stable use of large time steps, but efficient implementations of such methods require significantly more complex solvers that effectively treat both Lagrangian and Eulerian variables simultaneously. Several different approaches to solving the coupled Lagrangian-Eulerian equations have been proposed, but a complete understanding of this problem is still emerging. This paper presents a geometric multigrid method for an implicit-time discretization of the IB equations. This multigrid scheme uses a generalization of box relaxation that is shown to handle problems in which the physical stiffness of the structure is very large. Numerical examples are provided to illustrate the effectiveness and efficiency of the algorithms described herein. These tests show that using multigrid as a preconditioner for a Krylov method yields improvements in both robustness and efficiency as compared to using multigrid as a solver. They also demonstrate that with a time step 100--1000 times larger than that permitted by an explicit IB method, the multigrid-preconditioned implicit IB method is approximately 50--200 times more efficient than the explicit method.

The discontinuous Galerkin (DG) method is based on the idea of projection using integration. The recent direct flux reconstruction (DFR) method by Romero et al. (J Sci Comput 67(1):351–374, 2016) is derived via interpolation and results in a scheme identical to DG (on hexahedral meshes). The DFR method is further studied and developed here. Two proofs for its equivalence with the DG scheme considerably simpler than the original proof are presented. The first proof employs the $$ 2K - 1 $$ degree of precision by a $$ K $$-point Gauss quadrature. The second shows the equivalence of DG, FR, and DFR by using the property that the derivative of the degree $$ K + 1 $$ Lobatto polynomial vanishes at the $$ K $$ Gauss points. Fourier analysis for these schemes are presented using an approach more geometric compared with existing analytic approaches. The effects of nonuniform mesh and those of high-order mesh transformation (a precursor for curved meshes in two and three spatial dimensions) on stability and accuracy are examined. These nonstandard analyses are obtained via an in-depth study of the behavior of eigenvalues and eigenvectors.

Flow simulation of multimillion grid cell models with hundreds of wells and decades long production history can be extremely time-consuming. This often limits the applicability of assisted history matching techniques. A pragmatic solution to this problem is grid coarsening which is now embedded in many commercial reservoir simulators. Instead of up-scaling geological models in external packages, grid cells are automatically amalgamated within the simulator while preserving flux distribution and reducing the total number of active cells. The resulting speedup can be significant, often only at small loss of accuracy. Both characteristics are essential elements of any inversion technique in a multimillion grid cell environment. For water-flood history matching, we have utilized a commercial finite-volume simulator and the streamline-based generalized travel time inversion whereby water-cut behavior is matched by adjusting inter-well permeabilities. To apply the assisted history matching technique on high resolution models, a flux reconstruction method is devised which makes full use of the benefits of automatic grid coarsening. It approximates fluxes in the original geological grid by recalculating transmissibility at the fine scale and redistributing coarse scale fluxes accordingly. The inversion is thus conducted on the fine scale grid, while the forward simulation model uses the coarse grid. While solving for an inexpensive coarse model, the calibration takes into account fine grid resolution. Upscaling is not required and the degree of coarsening can be adjusted during the matching process. The proposed method was successfully tested on a supergiant carbonate oilfield with about hundred wells and large-scale water injection. The history match improved dramatically at relatively low numerical cost, which also allowed for investigating multiple sensitivities. The results were verified against non-coarsened model. The significant increase in efficiency makes this a potential method of choice for cases, where previously assisted history matching techniques could not be deployed due to excessive run times.

This study proposes the optimization of a low-level assembly code to reconstruct the flux for a splitting flux Harten–Lax–van Leer (SHLL) scheme on high-end graphic processing units. The proposed solver is implemented using the weighted essentially non-oscillatory reconstruction method to simulate compressible gas flows that are derived using an unsteady Euler equation. Instructions in the low-level assembly code, i.e. parallel thread execution and instruction set architecture in compute unified device architecture (CUDA), are used to optimize the CUDA kernel for the flux reconstruction method. The flux reconstruction method is a fifth-order one that is used to process the high-resolution intercell flux for achieving a highly localized scheme, such as the high-order implementation of SHLL scheme. Many benchmarking test cases including shock-tube and four-shock problems are demonstrated and compared. The results show that the reconstruction method is computationally very intensive and can achieve excellent performance up to 5183 GFLOP/s, ∼66% of peak performance of NVIDIA V100, using the low-level CUDA assembly code. The computational efficiency is twice the value as compared with the previous studies. The CUDA assembly code reduces 26.7% calculation and increases 37.5% bandwidth. The results show that the optimal kernel reaches up to 990 GB/s for the bandwidth. The overall efficiency of bandwidth and computation performance achieves 127% of the predicted performance based on the HBM2-memory roofline model estimated by Empirical Roofline Tool.

This dissertation, framed within an Industrial Doctorate programme, has the twofold objective of improving the understanding of numerical properties and convergence acceleration techniques for High-Order methods, and to formulate an efficient incompressible flow solver for thermal applications. The first part of this manuscript is devoted to the study of the linear multi-grid method applied to Flux Reconstruction, a relatively recent High-Order numerical method with close relations to the well-known Discontinuous Galerkin and Spectral Difference schemes. We do so by investigating the non-modal dissipation behaviour of the multi-grid scheme as a function of its parameters, in an effort to find the most efficient configurations. The analysis is extended to 2D high aspect-ratio cells, in the presence of dominating convection. The second part of this manuscript introduces a High-Order incompressible flow solver devised by the simultaneous use of the Flux Reconstruction, the Discontinuous Galerkin and the Continuous Galerkin methods, each of which is known to offer increased computational efficiency for certain types of problems. On the one end of the spectrum, the Flux Reconstruction is an efficient algorithm for the explicit evaluation of fluxes in conservative form; on the other end, the Continuous Galerkin method is comparatively very efficient for the resolution of diffusion-dominated steady problems. The different formulations are put together in a fractional-step approach and validated against known test cases – unsteady Stokes and Taylor problems, lid-driven cavity, Taylor-Green vortex and thermally-driven cavity – with and without analytical solution, demonstrating convergence with mesh, time-step and polynomial refinement. RESUMEN El presente trabajo, enmarcado en un programa de Doctorado Industrial, tiene el doble objetivo de mejorar la comprensión de las propiedades numéricas y técnicas de aceleración de convergencia para métodos de alto orden, así como de formular un solver eficiente para la resolución de flujos incompresibles con aplicaciones térmicas. La primera parte del manuscrito está dedicada al estudio del método multi-grid lineal aplicado a Flux Reconstruction, un método numérico de alto orden relativamente reciente y estrechamente relacionado con los conocidos Discontinuous Galerkin y Spectral Difference. Ello se consigue investigando la disipación numérica no-modal del esquema multi-grid en función de sus parámetros, con el objetivo de determinar aquellas configuraciones más eficientes. El análisis se extiende a 2D con elementos con alta relación de aspecto en presencia de convección dominante. La segunda parte del manuscrito presenta un solver de alto orden incompresible concebido con el uso simultáneo de los métodos Flux Reconstruction, Discontinuous Galerkin y Continuous Galerkin, cada uno de los cuales ofrece una eficiencia computacional superior para ciertos tipos de problemas. En un extremo, Flux Reconstruction es un algoritmo eficiente para la evaluación explícita de flujos en forma conservativa; en el otro extremo, Continuous Galerkin es comparativamente muy eficiente para la resolución de problemas elípticos en estado estacionario. Las diferentes formulaciones se conjugan en un método de paso fraccionado validado con conocidos casos test con y sin solución analítica, demostrando convergencia con el refinamiento en el mallado, el orden polinomial y el paso de tiempo.

A reservoir fluid flow simulator for the study of advanced subsurface processes is described. It is based on the volume balance formulation for implicitly determining the fluid pressure. A set of component conservation equations are used to determine the molar masses of each component, and an energy conservation law is used to determine the system emperature. Fluid property calculations are clearly separate from flow computations, and this is illustrated by a brief discussion on a compositional black-oil formulation. As well-flow is not of primary interest, only some very simple source and sink treatment has been implemented. Different degrees of implicitness have been tested. To try to keep the formulation simple, we have preferred to use a sequential approach in which the different equations are solved in sequence. Then for each equation, and implicit or explicit solver may be used. For the pressure, only implicit solvers have been used, while for the molar masses, explicit and different types of implicit discretization have been tried. However, we were not able to develop an implicit molar mass solver which yielded higher performance than a simple explicit solver. This appeared to be due to large decoupling errors and the increased time to solve the additional implicit equations. Consequently, the simulator is using an implicit pressure solver followed by a temperature and explicit molar mass solvers. Another aspect of the simulator is its use of general unstructured grids. On those grids, it is often necessary to apply a multi-point flux approximation scheme (MPFA). As many commercial simulators experience performance degradation using MPFA rather than a two-point flux approximation (TPFA), we provide some examples which show that in our case, MPFA is not considerably slower than TPFA on unstructured meshes. This is partly due to the choice of not using a high degree of implicitness, which reduces the size of the linear systems. Finally, the simulator parallelization strategy is outlined. A domain decomposition preconditioner is used as a solver, and the its performance is ascertained using a varying number of subdomains.

Hybridized formulations of flux reconstruction schemes for advection-diffusion problems

Parallel Computing Performance of an Implicit Gridless Type Solver

Reacting flow calculations are stiff for time-stepping. Recently, Zhong derived three sets of semi-implicit Runge-Kutta(SIRK) methods for split ordinary differential equations in the form of u' = /(«) 4g(u), where the nonstiff term / is treated explicitly while the stiff term g is simultaneously treated implicitly. The purpose of this work is twofold: first, to simulate multidimensional detonation waves using the high-order Rosenbrock semi-implicit Runge-Kutta time-stepping scheme(SIRK-3C); secondly to quantitatively compare the performance of SIRK with the second-order timesplitting schemes in dealing with the hyperbolic conservation laws with stiff relaxation terms. In the simulation of two-dimensional detonations, a full set of elementary hydrogen-oxygen kinetics with nine species are linear-implicitly treated while the explicit discretization of the basic advection equations are achieved by the third-order ENO schemes. Several model problems of hyperbolic systems with stiff relaxation are considered in the comparison of the Time-Splitting and the SIRK-3C scheme. A set of low-storage SIRK(LSSIRK) schemes is derived and tested to be high-order accurate and strongly A-stable. The results show that high-order SIRK time-stepping methods are suitable for stiff reactive flow simulations like the multi-dimensional detonations. INTRODUCTION In the reactive flow field solutions and the combustion related problems, the existence of several nonequilibrium states impose additional difficulties in the solutions of the reactive Euler equations with stiff relaxation terms. Since the smallest time scale is introduced from the chemical reaction kinetics, the second-order time-splitting method of Strang^ allows an independent calculation of the stiff ODE for the source term via implicit methods in one step, and an explicit highorder convective calculation in the following time steps. By the main source of stiffness from the basic hyperbolic conservation laws, it can accomplish a •Graduate Student, Member AIAA 'Assistant Professor, Mechanical and Aerospace Engineering Department, Member AIAA Copyright ©1997 by American Institute of Aeronautics and Astronautics, Inc. AH rights reserved. robust and easy-to-implement computation of reactive flow of current interests. Despite the second-order in time accuracy and the robustness of these methods, there are two main difficulties with the stiffness which have been minimally handled. The spurious solutions are one of the often observed phenomena, which are stable and free of oscillations, and yet may be completely incorrect.'-' Some type of resolution technique is needed to treat this known difficulties.^'!] The degrading of the second-order time accuracy to the first-order is another and quite recent observation when there exist solutions with many small-scaled structures of high-degree of complication. Jin® has modified the Strang splitting into the secondorder time-splitting methods which preserve its highorder accuracy when the considered solutions are complicated. New third-order semi-implicit Rosenbrock type Runge-Kutta scheme, based on the original version of Zhong'---', is computationally efficient with the linearized source term integration, thus requiring no iterations during the implicit treatment of the stiff source term. Third-order ENO schemes discretize the convective fluxes such that both implicit and explicit terms are treated at each of the three Runge-Kutta stages. This high-order Runge-Kutta scheme(SIRK-3C) still possesses the typical under-resolved characteristics of unphysical spurious numerical results of the TimeSplitting method. However the reduction to lower order if the small relaxation time is not temporally wellresolved, is successfully removed. Multi-dimensional Detonations Detonation waves are multi-dimensional and unstable phenomenon in nature as demonstrated by the early experiments of Urtiew and Oppenheinv-'. Existence of the triple points'^, consisting of an incident shock, a reflecting shock and a Mach stem, is the main characteristic of the reacting region behind the propagating shock front, and these detaching triple points from the leading front further contributes to the rolling up of vortices of opposite strength. One distinctive feature of the instability process observed in experiments is the formation of regular cell structures as triple points collide as they come together in the incident shock and move away from each other in the newly formed Mach stem. Figure[13] depicts this process of triple point col-